2Orthogonal polynomials
IB Numerical Analysis
2.1 Scalar product
The scalar products we are interested in would be generalization of the usual
scalar product on Euclidean space,
hx, yi =
n
X
i=1
x
i
y
i
.
We want to generalize this to vector spaces of functions and polynomials. We
will not provide a formal definition of vector spaces and scalar products on an
abstract vector space. Instead, we will just provide some examples of commonly
used ones.
Example.
(i)
Let
V
=
C
s
[
a, b
], where [
a, b
] is a finite interval and
s ≥
0. Pick a weight
function
w
(
x
)
∈ C
(
a, b
) such that
w
(
x
)
>
0 for all
x ∈
(
a, b
), and
w
is
integrable over [
a, b
]. In particular, we allow
w
to vanish at the end points,
or blow up mildly such that it is still integrable.
We then define the inner product to be
hf, gi =
Z
b
a
w(x)f(x)d(x) dx.
(ii)
We can allow [
a, b
] to be infinite, e.g. [0
, ∞
) or even (
−∞, ∞
), but we have
to be more careful. We first define
hf, gi =
Z
b
a
w(x)f(x)g(x) dx
as before, but we now need more conditions. We require that
R
b
a
w
(
x
)
x
n
d
x
to exist for all
n ≥
0, since we want to allow polynomials in our vector
space. For example,
w
(
x
) =
e
−x
on [0
, ∞
), works, or
w
(
x
) =
e
−x
2
on
(
−∞, ∞
). These are scalar products for
P
n
[
x
] for
n ≥
0, but we cannot
extend this definition to all smooth functions since they might blow up too
fast at infinity. We will not go into the technical details, since we are only
interested in polynomials, and knowing it works for polynomials suffices.
(iii) We can also have a discrete inner product, defined by
hf, gi =
m
X
j=1
w
j
f(ξ
j
)g(ξ
j
)
with
{ξ
j
}
m
j=1
distinct points and
{w
j
}
m
j=1
>
0. Now we have to restrict
ourselves a lot. This is a scalar product for
V
=
P
m−1
[
x
], but not for
higher degrees, since a scalar product should satisfy
hf, fi >
0 for
f 6
= 0.
In particular, we cannot extend this to all smooth functions.
With an inner product, we can define orthogonality.
Definition
(Orthogonalilty)
.
Given a vector space
V
and an inner product
h·, ·i, two vectors f, g ∈ V are orthogonal if hf, gi = 0.